Finite Group Databasebeta

← #28312453

Subgroups — #28312453

Order 13568.

398 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811534C212yesmaximal, complemented
2#20289852C212yesmaximal, complemented
3#20289850C212yesmaximal, complemented
4#20285175C211yesmaximal, characteristic
5C2×#13133278—22no—
6#16811534—22no—
7#16811534C2214yes—
8C2×#13133247—22no—
9C2×#13133217C2211yescharacteristic
10#16811018—22no—
11#16811002C411yescharacteristic
12#16811002C2211yescharacteristic
13#16810995C411yescharacteristic
14#16810995C2211yescharacteristic
15C4×#11719917C411yescharacteristic
16C53×#532C411yescharacteristic
17C2×#11719940—42no—
18#13133278—24no—
19#13133278—42no—
20C22×Dic106D412yes—
21#13133247—44no—
22#13133247—44no—
23C2×#11719925—22no—
24C2×#11719925—22no—
25C2×#11719923—22no—
26C2×#11719923—22no—
27C2×#11719917—21no—
28#13133217—41no—
29#13133217C2×C411yescharacteristic
30#13133217D412yes—
31#13133217C2311yescharacteristic
32C2×C4⋊Dic53—41no—
33#13133128—21no—
34C4×#88322C2×C411yescharacteristic
35C4×#88322C2×C411yescharacteristic
36C2×#41×C53—21no—
37C4⋊Q8×C53C2×C411yescharacteristic
38#127×C53C2×C411yescharacteristic
39#124×C53—21no—
40#121×C53—41no—
41C4×D4×C53—41no—
42C2×C42×C53C2×C411yescharacteristic
43#41×C53—42no—
44#41×C53—42no—
45C2×Q8×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C22×C4×C53—21no—
50C22×C4×C53—41no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×Dic53—22no—
53C2×Dic106—22no—
54C2×Dic106—42no—
55C2×Dic106—42no—
56C2×Dic106C22⋊C412yes—
57C2×Dic106C2×D412yes—
58#11719940—44no—
59#11719940—82no—
60C2×#88324—82no—
61C4⋊C4×C53—41no—
62C22⋊C4×C53—42no—
63C42×C53—21no—
64C42×C53C22⋊C411yescharacteristic
65C42×C53C22×C411yescharacteristic
66#11719925—44no—
67#11719925—44no—
68#11719923—44no—
69#11719923—44no—
70C4⋊Dic53—41no—
71C4⋊Dic53—42no—
72C4⋊Dic53—41no—
73#11719917—24no—
74C2×#88322—21no—
75C2×#88322—21no—
76C23×C53—41no—
77C23×C53#9911yescharacteristic
78Q8×C53—42no—
79D4×C53—42no—
80D4×C53—82no—
81D4×C53—82no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—42no—
86C2×C4×C53—41no—
87C2×C4×C53—81no—
88C2×C4×C53—81no—
89C2×C4×C53#9911yescharacteristic
90C2×C4×C53C2×C22⋊C411yescharacteristic
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95Dic106—42no—
96Dic106—42no—
97Dic106—82no—
98#88324—84no—
99#88322—22no—
100#88322—22no—
101#13590—531nomaximal, Sylow
102C22×C53—41no—
103C22×C53—82no—
104C22×C53—81no—
105C22×C53#35312yes—
106C22×C53C2×#9911yescharacteristic
107C4×C53—22no—
108C4×C53—21no—
109C4×C53—21no—
110C4×C53—41no—
111C4×C53—81no—
112Dic53—42no—
113Dic53—42no—
114#3152—531no—
115C2×#356—532no—
116#1411—532no—
117#1410—532no—
118C2×C53—21no—
119C2×C53—22no—
120C2×C53—161no—
121C2×C53C2×#35311yescharacteristic
122C2×#137—1062no—
123#532#615711yescharacteristic
124C2×C4⋊Q8—531no—
125#497—1062no—
126#494—531no—
127#494—531no—
128#492—531no—
129#492—531no—
130C2×#101—1062no—
131C4×M16—531no—
132#356—534no—
133#356—1062no—
134C53#1359011yescharacteristic, complemented, Sylow
135C2×#41—21no—
136C22×Q8—532no—
137#137—2124no—
138#137—2124no—
139C2×Q16—1062no—
140C2×Q16—1062no—
141C2×SD16—1062no—
142C2×SD16—1062no—
143C2×M16—1061no—
144C2×M16—2122no—
145C4⋊Q8—532no—
146C4⋊Q8—531no—
147C4⋊Q8—531no—
148C4⋊Q8—2121no—
149C4⋊Q8C2×#615711yescharacteristic
150#127C2×#615711yescharacteristic
151#124—21no—
152#121—41no—
153C4×D4—41no—
154C2×C4⋊C4—2121no—
155C2×C42C2×#615711yescharacteristic
156#101—1064no—
157#101—2122no—
158#97—1061no—
159C4×C8—531no—
160C4×C8—531no—
161#41—42no—
162#41—42no—
163C2×Q8—21no—
164C2×Q8—532no—
165C2×Q8—532no—
166C2×Q8—1062no—
167C2×Q8—2122no—
168C2×Q8—2122no—
169C2×D4—21no—
170C2×D4—41no—
171C2×D4—41no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—1062no—
175C22×C4C22⋊#615711yescharacteristic
176Q16—2124no—
177Q16—2124no—
178SD16—2124no—
179SD16—2124no—
180M16—1064no—
181M16—4242no—
182M16—2124no—
183C2×C8—1061no—
184C2×C8—1061no—
185C2×C8—4242no—
186C4⋊C4—41no—
187C4⋊C4—2121no—
188C4⋊C4—2121no—
189C4⋊C4—2122no—
190C22⋊C4—42no—
191C42—21no—
192C42C22⋊#615711yescharacteristic
193C42C22×#615711yescharacteristic
194C23—41no—
195C23#1313327511yescharacteristic
196Q8—42no—
197Q8—4242no—
198Q8—2122no—
199Q8—2122no—
200D4—42no—
201D4—82no—
202D4—82no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4#1313327511yescharacteristic
215C2×C4C2×C22⋊#615711yescharacteristic
216C8—1062no—
217C8—1062no—
218C8—4244no—
219C22—41no—
220C22—82no—
221C22—81no—
222C22#1681152712yes—
223C22C2×#1313327511yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—41no—
228C4—81no—
229C4—2122no—
230C4—2122no—
231C2—21no—
232C2—22no—
233C2—161no—
234C2C2×#1681152711yescharacteristic
subgroupG/H
centerC2C2×#16811527
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
Fitting, prime radicalC53×#532C4
socleC2×C53C2×#353
cosocle#13133217C23
hypercenter#532#6157
hypocenterC53#13590
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #532

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16811534C2semidirect
2. #20289852C2semidirect
3. #20289850C2semidirect
134. C53#13590semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168115342
#202898522
#202898502
#202851752
#1359053